The large, broad increase in wages is associated with the broad increase in the labor force. The Great Recession is associated with a decrease in wages and a negative shock to the labor force. The other four smaller wage increases occur just before recessions in a similar fashion to the smaller shocks impacting PCE inflation:

Essentially, this creates a picture where there are two kinds of shocks to wages: demographic and post-recovery/pre-recession shocks that occur just before recessions. There was a broad demographic increase in wages associated with women entering the workforce, and a smaller one associated with the Great Recession (I imagine early/forced retirements of some "Baby Boomers" [1]). The smaller and narrower positive shocks occur between recessions (centered in 1980 ‒ i.e. between the 1974 and 1981 recessions ‒ as well as 1989, 1997, and 2007). These match up with shocks to PCE inflation. This is not to say the post-recovery/pre-recession shocks cause recessions. They likely don't; what happens is that wages start to rise and a recession intervenes cutting the improvement short.

This would tell a story of why wages are stagnant: wages haven't increased because there haven't been any demographic increases in the labor force and because too many recessions have cut wage growth short. According to the model, wages grow at an average rate of 2.3%. However rising wages between recessions are a significant component of higher wages.

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Footnotes:

[1] This creates an interesting hypothesis: was the Great Recession bad simply because it occurred when Baby Boomers started to reach retirement age? The baby boom is generally associated with beginning in the 1940s, and the 2008 recession was the first one after 1940 + 65 years = 2005. Forget the Fed's missteps or over-leveraged banks ‒ was the Great Recession inevitable after the post WWII baby boom?